In the second instance it is a generalisation of the Hurwitz-Lerch zeta function, namely the higher-order Apostol-Lerch function defined by the integral
[1]}, [2]}, [3]}
\(\zeta (z,\sigma ,v,\lambda ) = \frac{1}{\Gamma (\sigma )} \int _{0}^{\infty }dx\; \frac{x^{\sigma -1}e^{-vx}}{\left( 1-z e^{-x}\right)^{\lambda }} ,\) 
